It is commonplace to consider applications of mathematics to other fields, especially the exact sciences. But what I would like to know about is the converse topic, namely:

What are some applications of other
fields to mathematics?

Obviously the applications of physics to mathematics are ubiquitous (gauge theory is just one significant modern example, and quantum algorithms and mirror symmetry are others...the list from physics goes on). For the purposes of this question (at least) theoretical computer science is just a branch of mathematics.

So answers involving fields other than physics are of particular interest to me (and answers involving theoretical computer science are of little to no interest to me), as are answers where the application isn't bidirectional (for example, one could say that game theory is an application of mathematics to economics as much if not more than an application of economics to mathematics).

Finally (at least for the purposes of this question), anything of the form "phenomenon Y was experimentally observed and it turned out that there was a rich but hitherto unknown mathematical theory Z explaining Y" is not that interesting as an application to mathematics unless the discovery of Z has some truly special status. Something like (e.g.) symplectic geometry might fall under this (leaving aside the "experimental" bit), but is not of particular interest for reasons above.

I have deleted a discussion in the comments about having too many "big-list" questions on the front page. Please follow the link above to meta if you're interested in that discussion.
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Scott Morrison♦Feb 9 '10 at 18:49

The Lotka-Volterra predator-prey equations are a fundamental example in the qualitative theory of ODEs. Volterra originally used it to explain the large increase in the mediterranean shark population during WWI.

I’d like to add something regarding game theory. Certainly, it is generally true that – as stated in the question - the application isn’t bidirectional.

However, there are exceptions. I heard of some contributions of the so called “Infinite games with perfect information” to the fields of:

mathematical logic and set theory – e.g. the Axiom of determinacy,

topology – topological games.

I think that especially the contribution to mathematical logic is a nice example of the ”unexpected inversion” (a term used by Scott Aaronson in his answer to this question). We have a situation when the applied leave (game theory) contributes to the root of the tree.

Gerry Myerson's answer about Gordan and theology was humorous, but Georg Cantor really did use theology in his conception of set theory. From Cantor's Wikipedia biography:

The concept of the existence of an
actual infinity was an important
shared concern within the realms of
mathematics, philosophy and religion.
Preserving the orthodoxy of the
relationship between God and
mathematics, although not in the same
form as held by his critics, was long
a concern of Cantor's.[51] He directly
addressed this intersection between
these disciplines in the introduction
to his Grundlagen einer allgemeinen
Mannigfaltigkeitslehre, where he
stressed the connection between his
view of the infinite and the
philosophical one.[52] To Cantor, his
mathematical views were intrinsically
linked to their philosophical and
theological implications—he identified
the Absolute Infinite with God,[53]
and he considered his work on
transfinite numbers to have been
directly communicated to him by God,
who had chosen Cantor to reveal them
to the world.[12]

Misha Gromov's recent Bull. AMS article "Crystals, proteins, stability and isoperimetry" (2011) can be read as a 29-page essay on the requested topic, with a focus particularly on mathematical inspiration arising in evolutionary biology, neurophysiology, and cognitive science. Gromov sets the stage as follows:

One may conjecture that neither cell nor brain would be possible if not for
profound mathematical “somethings” behind these, Nature’s inventions. But what
are these “somethings”? Why do we, mathematicians, remain unaware of them? … The history of mathematics shows how slow we are when it comes to inventing/recognizing new structures even if they are spread before our eyes, such as hyperbolic space, for instance. … One has to browse through myriad stars—structural specks of Life revealed by biologists—in order to identify the “essential ones”, and when (if ?) we ﬁnd them, we may start on the long road toward new mathematics.

Gromov then goes on to suggest many dozens of concrete questions, arising in many bio-related disciplines, that uniformly direct our vision (to use Scott Aaron's nice similes) from the "leaves" of life to the "roots" of fundamental mathematics.

What does Gromov see (that everyone sees) that inspires him so frequently to conceive mathematics (that no one previously has conceived (Szent-Gyrgyi))? Gromov has written this essay, to tell us precisely what it is, that he presently sees.

Since this is a community Wiki, I will informally suggest that it is great fun to read Gromov's inspiring essay either immediately before, or immediately after, viewing Stephane Guisard and Jose Salgado's similarly inspiring VLT (Very Large Telescope) HD Timelapse Footage.

Gromov is concerned largely with very small (molecular-scale) evolved systems, while Guisard and Salgado are concerned mainly with very large (galactic scale) evolved systems … and yet they are tapping the same source.

Not sure if this counts, but two rather effective global optimization algorithms of stochastic type were in fact inspired from work in different fields: "simulated annealing" rests on a physical analogy to the slow cooling (annealing) of metals, while "differential evolution" appeals to an analogy to mating of pairs and the subsequent mutation of their offspring.

I can think of at least three things that the question might mean, and it would probably help if Steve clarified which ones count for him!

(1) Other fields suggesting new questions for mathematicians to think about, or new conjectures for them to prove. Examples of that sort are ubiquitous, and account for a significant fraction of all of mathematics! (Archimedes, Newton, and Gauss all looked to physics for inspiration; many of the 20th-century greats looked to biology, economics, computer science, etc. Even for those mathematicians who take pride in taking as little inspiration as possible from the physical world, it's arguable how well they succeed at it.)

(2) Other fields helping the process of mathematical research. Computers are an obvious example, but I gather that this sort of application isn't what Steve has in mind.

(3) Other fields leading to new or better proofs, for theorems that mathematicians care about even independently of the other fields. This seems to me like the most interesting interpretation. But it raises an obvious question: if a field is leading to new proofs of important theorems, why shouldn't we call that field mathematics? One way out of this definitional morass is the following: normally, one thinks of mathematics as arranged in a tree, with logic and set theory at the root, "applied" fields like information theory or mathematical physics at the leaves, and everything else (algebra, analysis, geometry, topology) as trunks or branches. Definitions and results from the lower levels get used at the higher levels, but not vice versa. From this perspective, what the question is really asking for is examples of "unexpected inversions," where ideas from higher in the tree (and specifically, from the "applied" leaves) are used to prove theorems lower in the tree.

Such inversions certainly exist, and lots of people probably have favorite examples of them --- so it does seem like great fodder for a "big list" question. At the risk of violating Steve's "no theoretical computer science" rule, here are some of my personal favorites:

There's a connection between random walks and electric networks (the link goes to the book of that title by Doyle and Snell); this is "physics", I suppose, but hopefully not the sort you meant to exclude!

This blog post of Kenny Easwaran suggests a mathematical result that can be explained using economic intuition. It's more an interesting isolated example than a genuine application, but it's still interesting.

Chaitin describes in his book "Meta Math! The Quest for Omega" his point of view on information theory, complexity theory and a number of other questions (some of which should not be taken too serious, e.g. when he's comparing evolution and quantum physics). The whole book is dedicated to telling the story of how he discovered whta is now called Chaitin's constant and the theory connect with it by thinking about rather simple(looking) questions in computer sciences. (It is therefore a very unusual math book)

If someone is interested in how non-math-question can lead to new math, this book is definitely one of the places to find examples. You can find a online reader for the book here.

As a contribution to mathematical practice (as opposed to mathematics itself), you might consider the emerging field of mathematical cognition (also known as cognitive science of mathematics), i.e. the study of mathematical ideas and their empirical grounding in human experiences, metaphors, generalizations, analogies and other cognitive mechanisms.

This subject has been explored informally and non-rigorously by mathematicians such as Saunders Mac Lane (see his Mathematics, Form and Function), but until recently it was not pursued by researchers trained in cognitive science. The best-known introduction is the book Where Mathematics Comes From by G. Lakoff and R. Nunez, but research has also been undertaken by Brian Rotman (Mathematics as sign: Writing, imagining, counting) and others.

Mathematics need some source of new ideas, and any other fields are very good sources of ideas: art, philosophy, physics, music, even sport probably.
From the other side may important discoveries are made by means not-so-precise formulations of solutions known problems. Feynman path integral is good example, distributions theory as well etc. Forgive me, but math without such external source of ideas, without any contact with reality ( not exactly direct contact) in my opinion is not very productive nor interesting.

I can describe one that is still a little mysterious to me. My colleagues Erwin Lutwak and Gaoyong Zhang and I have shown how ideas arising from the continuous version of Shannon information theory (which normally resides in the electrical engineering department) lead very naturally to sharp analytic inequalities for functions on $R^n$, including generalized sharp Sobolev inequalities. What's really nice is that not only does this point of view lead to the inequalities, it also leads to much nicer and easier to understand proofs of the inequalities than previously known proofs.

It is perhaps a bit of a historical fluke that information theory lives in EE. This, of course, stems from its roots in communication theory. Pure information theory might be better characterized as a branch of statistics, or possibly even physics. That said, another good example along these lines is the paper Information theoretic inequalities by Dembo, Cover & Thomas, where, for instance, some classical determinantal inequalities are derived from consideration of entropies of multivariate normals.
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Jon YardFeb 10 '10 at 1:18

1

@Jon: You make a good point, and I really should have cited the work of Dembo, Cover, and Thomas, as well as other people. They have done some beautiful mathematics disguised as EE, and I think there is a lot more to be explored.
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Deane YangFeb 10 '10 at 1:56

2

@Jon Yard, @Deane Yang, wasn't there a turf battle between mathematics and electrical engineering for "ownership" or "rights" to computer science and information theory. At some places, computer science is a part or division of Electrical Engineering. At few, computer science is a branch of mathematics. And at many places, it is a department in its own right. The same thing happens with Neurosurgery. Some schools have Divisions of Neurosurgery as part of the Department of Surgery, others free-standing Depts of Neurosurgery, and for some odd ducks it's part of Neurology, a medical branch.
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ApurvaSep 16 '10 at 22:19

6

There is a lot of evidence that many EE and CS professors are really mathematicians, even pure mathematicians, who figured out that life would be much better (higher salaries, lower teaching load, better grants) if they disguised themselves as working in EE or CS. There is a lot of extremely deep mathematics being done in the name of EE or CS.
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Deane YangSep 17 '10 at 1:58